The face pair of planar graphs

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24–26] have introduced a generalization of the standard cage question—r  -regular graphs with given odd and even girth pair. The pair (ω,ε)(ω,ε) is the girth pair of graph G if the shortest odd and even cycles of G   have lengths ωω and εε, respectively, and denote the number of vertices in the (r,ω,ε)(r,ω,ε)-cage by f(r,ω,ε)f(r,ω,ε). Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145–153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f(3,ω,4)=2ωf(3,ω,4)=2ω and the bounds for the left cases. In this paper, we show the values of f(r,ω,ε)f(r,ω,ε) for the left cases where (r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε)(r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε), (3,5,ε)}(3,5,ε)}.